This text is concerned with those aspects of mathematics that are necessary for first-degree students of chemistry. It is written from the point of view that an element of mathematical rigour is essential for a proper appreciation of the scope and limitations of mathematical methods, and that the connection between physical principles and their mathematical formulation requires at least as much study as the mathematical principles themselves. It is written with chemistry students particularly in mind because that subject provides a point of view that differs in some respects from that of students of other scientific disciplines. Chemists in particular need insight into three dimensional geometry and an appreciation of problems involving many variables. It is also a subject that draws particular benefit from having available two rigorous disciplines, those of mathematics and of thermodynamics. The benefit of rigour is that it provides a degree of certainty which is valuable in a subject of such complexity as is provided by the behaviour of real chemical systems. As an experimen tal science, we attempt in chemistry to understand and to predict behaviour by combining precise experimental measurement with such rigorous theory as may be at the time available; these seldom provide a complete picture but do enable areas of uncertainty to be identified.
|Edition description:||Softcover reprint of the original 1st ed. 1984|
|Product dimensions:||5.51(w) x 8.50(h) x 0.02(d)|
Table of Contents1 Algebraic and geometrical methods.- 1.1 Natural numbers.- 1.2 Units and dimensional analysis.- 1.3 Functional notation.- 1.4 Quadratic and higher-order equations.- 1.5 Dependent and independent variables.- 1.6 Graphical methods.- 1.7 Some geometrical methods.- 1.7.1 Similar triangles.- 1.7.2 Triangular graph paper.- 1.7.3 Three-dimensional geometry.- 1.7.4 Circle, ellipse, parabola and hyperbola.- 1.7.5 Plane polar coordinates.- 1.8 Factorials and gamma functions.- 1.9 Probability.- 1.10 Complex numbers.- 2 Differential calculus.- 2.1 Significance and notation.- 2.2 The calculus limit.- 2.3 Differentiation of simple functions.- 2.4 The use of differentials; implicit differentiation.- 2.5 Logarithms and exponentials.- 2.6 The chain rule and differentiation by substitution.- 2.7 Turning points: maxima, minima and points of inflection.- 2.8 Maxima and minima subject to constraint; Lagrange’s method of undetermined multipliers.- 2.9 Series.- 2.9.1 Geometric series.- 2.9.2 Power series and Taylor’s theorem.- 2.9.3 Maclaurin’s theorem.- 2.9.4 Inversion of a series.- 2.9.5 Empirical curve fitting by power series.- 2.10 The evaluation of limits by L’Hôpital’s rule.- 2.11 The principles of Newtonian mechanics.- 3 Differential calculus in three or more dimensions; partial differentiation.- 3.1 Significance and notation.- 3.2 An alternative approach to calculus.- 3.3 The total differential.- 3.4 General expression for a total differential.- 3.5 Exact differentials.- 3.6 Relations between partial derivatives.- 3.7 Extensive and intensive variables; Euler’s theorem.- 3.8 Taylor’s theorem in partial derivatives.- 3.9 Vectors.- 3.9.1 Scalar and vector products.- 3.9.2 Scalar and vector fields.- 4 Integration.- 4.1 Significance and notation.- 4.2 Standard methods of integration.- 4.2.1 Simple functions.- 4.2.2 Reciprocal functions.- 4.2.3 Integration by parts.- 4.2.4 Integration by substitution.- 4.2.5 Expansion of algebraic functions.- 4.2.6 Integration by partial fractions.- 4.3 Standard forms of integral and numerical methods.- 4.4 Multiple integration.- 4.5 Differentiation of integrals; Leibnitz’s theorem.- 4.6 The EulerMaclaurin Theorem.- 5 Applications of integration.- 5.1 Plane area.- 5.2 Plane elements of area.- 5.3 Elements of volume; polar coordinates in three dimensions.- 5.4 Line integrals.- 5.5 Curve length by integration.- 5.6 Applications of multiple integration.- 5.6.1 The pressure of a perfect gas.- 5.6.2 Interactions in a real fluid.- 5.7 The calculus of variations.- 5.8 Generalized dynamics.- 6 Differential equations.- 6.1 Significance and notation.- 6.2 Equations of first order, first degree.- 6.2.1 Separable variables.- 6.2.2 First-order homogeneous equations.- 6.2.3 Exact equations.- 6.2.4 Linear equations of first order.- 6.3 Linear differential equations.- 6.3.1 Homogeneous linear equations with constant coefficients.- 6.3.2 General linear equation with constant coefficients.- 6.3.3 Linear equations of second order.- 6.4 Integral transforms.- 6.4.1 Laplace transforms.- 6.4.2 Fourier transforms.- 7 Experimental error and the method of least squares.- 7.1 Significance.- 7.2 Root-mean-square error.- 7.3 Distribution of error.- 7.4 The statistical analysis of experimental data.- 7.5 Propagation of error.- 7.6 Small-sample errors.- 7.7 The normal distribution of error.- 7.8 The method of least squares.- 7.8.1 Linear relation between two variables.- 7.8.2 Covariance and correlation coefficient.- 7.8.3 Uncertainty in the slope and intercept of the least-squares straight line.- 7.8.4 Least-squares straight line with both variables subject to error.- 7.8.5 Weighting of observations.- 7.8.6 Multivariable and non-linear least-squares analysis.- Appendix SI units, physical constants and conversion factors; the Greek alphabet and a summary of useful relations.- Index..